Graphing a quadratic
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Graphing a Quadratic. Warm-up. Determine the following algebraically (no calculator) vertex x- and y- intercepts . Is the vertex a max or min? How would you know without graphing? Identify intervals of increasing/decreasing. Write f in standard form (complete the square).

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Graphing a Quadratic

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Graphing a quadratic

Graphing a Quadratic

Warm-up

  • Determine the following algebraically (no calculator)

    • vertex

    • x- and y-intercepts.

    • Is the vertex a max or min? How would you know without graphing?

    • Identify intervals of increasing/decreasing.

  • Write f in standard form (complete the square)


Graphing a quadratic

Section 3.2 Polynomial Functions and Models

  • Polynomial Functions and their Degree

  • Properties of Power Functions

  • Zeros of a Polynomial and Multiplicity

  • Building Polynomials

  • End Behavior of Polynomials

    • Leading Coefficient Test

  • Analyzing the Graph of a Polynomial Function


1 polynomial functions and their degree

1. Polynomial Functions and their degree

A polynomial function is a function of the form

where the coefficients are real numbers

and n is a nonnegative integer.

The degree of the function is the largest power of x.

Factored form: Add the degrees


Polynomial functions and their degree

Polynomial Functions and their degree

Determine which of the following are polynomial functions. For those that are, state the degree.

1.

2.

3.

4.

5.


Polynomial functions

Polynomial Functions

Determine which of the following are polynomial functions. For those that are, state the degree.

6.

7.

8.

9.

10.


Polynomial functions and their degree1

Polynomial Functions and their degree

Determine which of the following are polynomial functions. For those that are, state the degree.

T; 56. T; 3

2. F7. T; 7

3. T; 58. T; 3

4. F9. F

5. T; 010. F


2 properties of the power functions

2. Properties of the Power Functions

A power function is of the form

If n is odd integer

If n is even integer

Symmetry:

Domain:

Range:

Key Points:


3 zeroes of polynomials and their multiplicity

3. Zeroes of Polynomials and their Multiplicity

  • Factor TheoremIf is a polynomial function, the following are equivalent statements:

    • r is a real number for which:

    • r is called a zero or root of

    • r is an x-intercept of the graph of

    • is a factor of

Example: Factor

For each factor, summarize properties 1-4 of the Factor theorem..


3 zeroes of polynomials and their multiplicity1

3. Zeroes of Polynomials and their Multiplicity

Definition: The multiplicity of a zero is the degree of the factor

Notation:

If f has a factor we say is a zero of multiplicity

Example: Identify the zeros and their multiplicities of:

Degree of f =

Graph f(x).


3 zeroes of polynomials and their multiplicity2

3. Zeroes of Polynomials and their Multiplicity

Example – continued.

What does the value of m tell us about the graph ?


3 a large values of multiplicity

3. a) Large values of Multiplicity

Analyze the graph of

What happens at the zero as m gets large?

Use your graphing calculator to graph the following:

1)

2)

The graph flattens out at the zero as the multiplicity increases.


3 more practice

3. More Practice

State the degree of this polynomial.

How many zeros does this function have?


4 building polynomials

4. Building Polynomials

Given that f has zeros with multiplicity

we can write:

Write a polynomial with these properties:

1) Degree 4: Zeroes at: -5, -4, 0, 2, (multiplicity 1)

2) Degree 3: Zeroes at: -3, multiplicity 2;

5, multiplicity 1

and passing through the point (0,9)


4 building polynomials1

4. Building Polynomials

Construct a polynomial function that might have this graph.


5 end behavior of polynomials

5. End Behavior of Polynomials

The end behavior of the graph is determined by the coefficient and degree of highest degree term

Sketch the graph of these functions. What is the end behavior?


5 a leading coefficient test

5. a) Leading Coefficient Test

Leading term determines end behavior

1. For n even.

2. For n odd.

Rises to left

and falls to right

Rises to left

and rises to right

Falls to left

and falls to right

Falls to left

and rises to right


6 analyzing the graph of a polynomial function

6. Analyzing the Graph of a Polynomial Function

Given the polynomial

What is the degree of this polynomial?

1. End behavior.

2. x-intercepts. Solve f(x) = 0

Behavior at each intercept (even/odd)

b) If k > 1, graph flattens for larger values of k.

3. y-intercepts. Find f(0).

4. Symmetry: Odd/Even

5. Turning points: Graph changes between increasing/decreasing.


Zeroes of polynomials

Zeroes of Polynomials


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